toolbox_chaos v0.1.0
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Toolbox Chaos practical guide

Compare Integration Methods

Run Euler, Heun, and RK4 on a compatible continuous ODE flow under the same inputs, interpret finite-horizon separation, and design a defensible step-refinement check.

Level Intermediate
GUI tab Comparar métodos
Research question How dependent is the computed trajectory on the chosen fixed-step integrator and step size over the declared horizon?

Objective

What you will accomplish

Evaluate numerical sensitivity for a compatible ODE flow without confusing solver disagreement with a newly discovered dynamical effect.

Before you begin

  • Use a compatible continuous ODE flow, preferably a supported three-state catalog case such as Lorenz, Rossler, or Chua, with a fixed parameter vector, initial condition, step, and duration.
  • Do not use this tutorial for discrete maps, Mackey–Glass delay dynamics, or Lorenz-96; their update contracts are not an effective Euler/Heun/RK4 comparison in this panel.
  • Understand that sensitive trajectories can diverge pointwise even when two reliable methods reproduce the same statistical regime.

Reference outputs

What these views can show

Current Comparar métodos tab overlaying Euler, Heun, and RK4 for Lorenz
The current panel applies one common model setup to every selected fixed-step method and displays the state overlays.
Reference Lorenz trajectory and state signals for a numerical comparison
Use a known reference setup to define which geometric and temporal features the method comparison should preserve.

Procedure

Step-by-step workflow

  1. Define a common experiment

    Select one compatible continuous ODE flow, its parameters, one initial state, step dt, and duration T. These values are shared by all checked methods.

    Begin with a moderate duration so early error growth can be read before long-time sensitive divergence dominates.

  2. Select the methods

    Enable Euler explícito, Heun / Euler mejorado, and Runge–Kutta 4. Assign distinct high-contrast colors.

    All three are fixed-step methods in this interface; the comparison does not include adaptive error-controlled integration.

  3. Generate the overlay

    Press Comparar integradores and inspect early agreement, phase drift, amplitude changes, boundedness, and any method-specific failure.

    Do not infer that the most visually complex trace is the most accurate one.

  4. Repeat at half the step

    Keep model, initial state, and horizon fixed; halve dt and regenerate the overlay.

    Look for improved early agreement and convergence of bounded regions, dominant frequencies, extrema distributions, or other quantities relevant to the study.

  5. Separate transient and asymptotic questions

    Use a short window to assess local integration agreement and a longer window to compare robust summaries.

    If a conclusion changes when halving the step, treat it as numerically unresolved rather than selecting the preferred-looking run.

  6. Export a comparison record

    Save the comparison figure for each tested step and name the files with the step and horizon.

    Report the list of enabled methods; do not write only ‘methods compared’ because unchecked methods are not part of the result.

A practical comparison matrix

  • Rows: Euler, Heun, and RK4.
  • Columns: dt, dt/2, and—when needed—dt/4.
  • Fixed inputs: equations, parameter vector, initial condition, and observation horizon.
  • Declared outputs: early-time deviation, boundedness, phase portrait, dominant spectral peaks, or another problem-specific summary.
  • Decision: adequate, inadequate, or unresolved at the tested resolution.

Result

Expected output

  • An overlay showing where Euler, Heun, and RK4 agree or separate for identical inputs on a compatible ODE flow.
  • A two-level step study that indicates whether the qualitative conclusion is stable enough for exploratory use.

Interpretation

How to read it

Early-time agreement that improves with step refinement supports numerical consistency. Late-time separation can be expected in a sensitive system and should not be presented alone as method failure.

A solver that diverges, changes the apparent regime, or retains large discrepancies after refinement signals that the chosen step or method is inadequate for the stated purpose.

Export

Reproducibility checklist

  • Record every enabled method, dt, T, parameters, and initial state.
  • Export both original-step and refined-step figures.
  • State the comparison criterion: early trajectory, geometry, extrema, spectrum, or another declared quantity.

Applications

Where this workflow helps

  • Teaching truncation error and numerical sensitivity.
  • Selecting a practical method and step before a larger parameter campaign.
  • Documenting that a qualitative portrait or diagnostic is not a coarse-step artifact.

Limits

What it does not establish

  • The panel compares implemented fixed-step methods; it is not an error estimator or adaptive integrator benchmark.
  • This workflow is limited to compatible continuous ODE flows. It does not provide an effective three-integrator comparison for maps, delay equations, or Lorenz-96.
  • Pointwise long-time agreement is generally unsuitable as the only criterion for chaotic trajectories.
  • Method agreement does not prove chaos, attraction, or hiddenness.